Secure and private authentication protocol is important in Radio Frequency Identification (RFID) technology. To date, researchers have proposed many RFID authentication protocols. However, these protocols have many flaws due to lack of theoretical support in designing these protocols. In this work, first we present the security and privacy requirements in RFID authentication protocols. Then we examine related works and point out problems in designing RFID authentication protocols. To solve these problems, we propose,and briefly prove three theorems. We also give necessary examples for better understanding these theorems with concrete protocols. At last, we give our suggestions on designing secure and private authentication protocols. The security and privacy requirements, theorems, and suggestions will facilitate better understanding and designing of RFID authentication protocols in the future.